Rotation and Magentic Fields Flashcards Preview

PHYS3281 Star and Planet Formation > Rotation and Magentic Fields > Flashcards

Flashcards in Rotation and Magentic Fields Deck (18)
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1
Q

Angular Momentum

Definition

A

L = mvr = mωr² = Iω
-where I = inertia is given by:
I = ∫ r²dm, ω=dθ/dt

2
Q

Cloud Rotation

Description

A

-conservation of specific angular momentum along the equator means that:
ωr² = ΩR²
-where R is the radius of the cloud and Ω is the angular momentum at the surface of the cloud
-and ω is the angular momentum at radius r within the cloud

3
Q

Cloud Rotation

Centrifugal Force

A

Fc = mω²r ∝ r^(-3)

4
Q

Cloud Rotation

Gravitational Force

A

Fg = GMm/r² ∝ r^(-2)

5
Q

Cloud Rotation

Balancing of Forces

A

-centrifugal force will eventually win out over the gravitational force and halt collapse along the equatorial plane where it is largest

6
Q

Cloud Rotation

Centrifugal Radius

A

-the radius at which centrifugal and gravitational forces are balanced:
Rc = Ω²R^4 / GM
-typically 100-1000au

7
Q

Typical Angular Velocity for Surface of Molecular Clouds

A

Ω = 10^(-14) rad/s

8
Q

How can material continue to infall if a point is reached where centrifugal and gravitational forces balance?

A

-the balancing of centrifugal and gravitational forces only considers how rotation at the equator impedes inward motion, the cloud can still contract parallel to the rotation axis leading to a flattened structure
-the rotational velocity decreases with increasing latitude:
vo = ωrosin(θo)
-the centrifugal force decreases and only acts perpendicular to the roation axis, material can continue to infall

9
Q

Cloud Rotation

Disc-Like Geometry Formation

A
  • infalling material flows along streamlines
  • there is a pile up of material at the centrifugal radius, rc
  • material continues to infall to the centre from other angles
  • the resulting density distribution shows the material concentrated in a disc-like geometry, with rdisk~2*Rc
10
Q

The Angular Momentum Problem

A

-in order to conserve angular momentum during collapse, the rotational frequency must increase:
ωf = (ri/rf)²*ωi OR vf = ri/rf * vi
-the collapse of even very slowly rotating clouds will result in a massive amplification of the rotational velocity
-for a typical cloud this gives a final rotational frequency of 0.1rad/s and a final rotational velocity of 10^8 m/s
-these are relativistic speeds, much faster than we observe stars to move
-the angular momentum must be transferred elsewhere, i.e. it is no conserved within the cloud during collapse

11
Q

How does a molecular cloud lose angular momentum during collapse?

A
  • fragmentation/fission, transfer of angular momentum to a cluster or a binary or planets
  • transfer of angular momentum through cloud-cloud interactions
  • magnetic braking of the star, charged particles couple with the magnetic field and resist angular motion
  • mass loss through outflows, the mass lost carries with it angular momentum
12
Q

Angular Momentum Transfer

Magnetic Breaking of a Star

A
  • the cloud exists in a charged plasma
  • this charge fluid velocity bends the magnetic field and creates a resisting tension force
  • any spin up during collapse twists the field and increases the local magnetic tension, this tension creates a braking torque on the element that counter acts the spin up and lowers the specific angular momentum
13
Q

How can angular momentum be transferred locally?

A
  • for this to happen we need a force linking the rapidly rotating inner core to the slowly rotating outer cloud
  • there are two possibilities:
  • -turbulent viscosity
  • -magnetic fields
14
Q

Local Angular Momentum Transfer

Turbulent Viscosity

A
  • friction between neighbouring annuli will create torques that act to bring the annuli into co-rotation
  • the outer annulus will try to speed up (i.e. gather angular momentum) angular momentum is transferred outwards
  • an exchange of particles bringing differing angular momenta to the annuli to which they are transferred
15
Q

Local Angular Momentum Transfer

Magnetic Fields

A
  • friction between neighbouring annuli will create torques that act to bring the annuli into co-rotation
  • the outer annulus will try to speed up (i.e. gather angular momentum) angular momentum is transferred outwards
  • inner and outer annuli are attached by ‘elastic spring’ which becomes stretched due to shear creating a restoring force (magnetic braking)
16
Q

The Magnetic Flux Problem

A

-molecular clouds are threaded with magnetic fields
Fb ∝ R²B²
-charged particles spiral around field lines effectively freezing the magnetic field into the material
-if the magnetic field is strong and uniform we can expect flattened clouds, as the material cannot cross the field lines, it can only move along them
-because the field lines are frozen into the material, we expect conservation of magnetic flux: ϕ∝BR²
-since both Fb and Fg are ∝R^(-2), magnetic force cannot overcome gravitational force once collapse has started
-conservation of magnetic flux gives relation:
B* R² = Bc Rc²
-giving a typical star magnetic field:
B
= 10^3 T
-this is significantly higher than observed, the system must lose magnetic flux at some point

17
Q

How is magnetic flux lost?

A

-ambipolar diffusion

18
Q

Ambipolar Diffusion

A
  • magnetic fields can only act on the ions/elecrons, not the neutral particles in the cloud
  • neutrals can drift relative to the magnetic field opposed only by collisions with ions, i.e. friction slows down the neutrals
  • slowly a cloud supported by a B field will expel the field and contract
  • eventually it will no longer be able to support itself and collapse
  • the timescale for ambipolar diffusion is typically longer than the free-fall timescale so it must occur before collapse